Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set $F\subseteq I$ $$\mu\Bigg(\bigcap_{i\in F}\pi^{-1}(A_i)\Bigg)=\prod_{i\in F}\mu_i(A_i),$$ when $A_i\in \Sigma_i$ for all $i\in F$.
The first valid proof of this theorem in full generality (without topological assumptions) seems to be due to Shizuo Kakutani in 1943 in Notes on infinite product measure spaces, I. Kakutani mentions in footnote 3 that a proof was attempted already in the 1934 paper Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités. I. Variables indépendantes by Łomnicki and Ulam, but that there was mistake in their argument. This is echoed in the historical remarks of Bogachev's encyclopedic book on measure theory. The following question is mainly motivated by historical curiosity:

What is, in rough outline, the proof
approach of Łomnicki and Ulam? What is
their mistake?

I'm sorry that my ignorance of French keeps me from answering the question by simply studying the original paper.